Third party peripherals can be problematic: mechanically, driver code, and documentation. Difficult to truly and really know what may or may not work.
And some product/unknown brand manufacturers are not (as you have learned) at all what is expected.
I looked for and found the website for the "Lizard".
https://en.tracer.eu/gamepad-tracer-lizard-pc/ps3/xinput
Found the User Manual:
https://en.tracer.eu/gfx-base/files/TRA/JOY/45206/inst_45206.pdf
English (Page 2).
The question is, I think, not so much a mechanical matter per se. When you squeeze/pull the trigger some signal is sent to the drivers which in turn tell the software application/game what is to do be done.
As I understand the manual the gamepad has configurable modes to match the game host: PC, Xbox.
Take a look at the Lizard manual and do the same for the Trust GXT. Determine if the available options and configurations make sense or otherwise fit in with your gaming plans.
If one or the other gamepad appears to be a suitable choice that is good. However if you have any questions go to the manufacturer's website and visit the Support pages. FAQs, Forums, Chat, etc.. Read what problems and questions other people have with respect to the gamepad in question. Post your own questions. See how (and when) the questions are answered. Or not.
And pay attention to what is said and not said. Vague responses are a warning.
Try downloading the drivers even though you do not have the specific device for the drivers. Connect a joystick or some other gamepad controller.
Do the driver test via Section 3.2 of the manual. The buttons may not do exactly what you wish but at least you will get a sense of the buttons and what happens when any given button is pressed.
Hopefully you will discover no problems or a problem that can be resolved by a configuration change. No certainty about any of it but you will most likely learn enough determine what will work and what will not work. Takes some effort.